On Tight Euclidean 6-Designs: An Experimental Result
Djoko Suprijanto
Combinatorial Mathematics Research Group Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung 40132, INDONESIA. Email: djoko@math.itb.ac.id
Abstract. A finite set $X \seq \RR^n$ with a weight function $w : X \longrightarrow \RR_{>0}$ is called \emph{Euclidean $t$-design} in $\RR^n$ (supported by $p$ concentric spheres) if the following condition holds: \[ \sum_{i="1"}^p \frac{w(X_i)}{|S_i|}\int_{S_i} f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x \in X}w(\boldsymbol x) f(\boldsymbol x), \] for any polynomial $f(\boldsymbol x) \in \mbox{Pol}(\RR^n)$ of degree at most $t$. Here $S_i \seq \RR^n$ is a sphere of radius $r_i \geq 0,$ $X_i=X \cap S_i,$ and $\sigma_i(\boldsymbol x)$ is an $O(n)$-invariant measure on $S_i$ such that $|S_i|=r_i^{n-1}|S^{n-1}|$, with $|S_i|$ is the surface area of $S_i$ and $|S^{n-1}|$ is a surface area of the unit sphere in $\RR^n$. Recently, Bajnok (2006) constructed tight Euclidean $t$-designs in the plane ($n="2"$) for arbitrary $t$ and $p.$ In this paper we show that for case $t="6"$ and $p="2",$ tight Euclidean $6$-designs constructed by Bajnok is the unique configuration in $\RR^n$, for $2 \leq n \leq 8.$
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